USING CONTINUOUS-TIME MODELING FOR ERRORS-IN-VARIABLES IDENTIFICATION

We review some applications of continuous-time ARMA processes in the modelling of time series observed at discrete times t 1,…, t N . The problem of finding a continuous-time ARMA process in which a given discrete-time ARMA process can be “embedded” is discussed and some of the properties of a family of continuous-time analogues of Tong’s SETARMA models are considered. An approach to inference for such models is described and illustrated with reference to a variety of data sets.KeywordsAutocorrelation FunctionStochastic Differential EquationTime Series AnalysisARMA ModelTransition DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.

On continuous-time threshold ARMA processes

This article deals with different approaches to continuous-time system identification from sampled data. Continuous-time system identification is important problem in control theory. Continuous time models provide many advantages against discrete time models because of better physical insight into the system properties. The traditional approach with least squares method with state variable filters is presented. Two alternative approaches to continuous-time identification are proposed. The generalized Laguerre functions method and the method based on least squares estimation with numerical solution of differential equation are introduced. These three different approaches to continuous-time system identification from sampled data are compared on the example. It is shown that proposed alternative methods can give better results in terms of relative root mean square error of the outputs of the identified systems than the least squares method with state variable filters.

Empirical spectral processes for stationary state space models

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

Identification of a MIMO continuous-time transfer function model with different denominators

This paper presents and demonstrates a general approach to solving spatial dynamic models in continuous space and continuous time that characterize the behaviour of intertemporally and interspatially optimizing agents and estimating from discrete data the parameters of such models. The approach involves the use of a projection method to solve the models and a quasi-Newton algorithm to update quasi-FIML parameter estimates.

Continuous-time modeling offers a flexible approach to analyze longitudinal data from designs with unequally spaced measurement occasions. Measurement models are popular tools in psychological research to control for measurement error. The objective of the present article is to introduce the continuous-time Rasch model, a combination of the Rasch model and a continuous-time dynamic model. In a series of simulations we demonstrate the performance of the proposed model and that ignoring individual unequal time interval lengths, choosing a wrong measurement model, and selecting a wrong analysis strategy results in poor parameter estimates. The newly proposed continuous-time Rasch model overcomes these problems and offers a powerful new approach to longitudinal analysis with dichotomous items. A step-by-step tutorial on how to run a continuous-time Rasch model with the R package ctsem and an illustrative empirical example is given. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Computation of time probability distributions for the occurrence of uncertain future events

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Levy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

Most models of income dynamics are set in a discrete-time framework with an arbitrarily chosen accounting period. This article introduces a continuous-time stochastic model of income flows, without the need to define an accounting period. Our model can be estimated using unbalanced panel data with arbitrarily spaced observations. Although our model describes the stochastic properties of income flows, estimation is based on observed incomes accruing during time intervals of possibly varying length. Our model of income dynamics is close in spirit to the discrete-time two-stage models prevalent in the literature. We impose a parsimoniously parameterized continuous-time stochastic process (possibly containing a unit root) to model the deviation from a traditional earnings function. We illustrate our approach by estimating a simplified model using microeconomic data from the German social security agency from 1975 to 1995.

Recently, an iterative algorithm has been presented for estimating the parameters of partially observed continuous-time processes [1]. In this note we concentrate on continuous-time ARMA processes observed in white noise. A maximum a-posteriori (MAP) estimator is defined for the trajectory of the parameters' random process. This approach enables the MAP estimation of randomly slowly varying parameters, and extends the conventional treatment of time-invariant parameters. The iterative algorithm derived for the MAP estimation, increases the posterior probability of the parameters in each iteration, and converges to a stationary point of the posterior probability functional. Each iteration involves a standard linear smoother followed by a finite-dimensional linear system, and thus is easily implemented.

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

For most continuous time models formulated in finance, there is no closed form for the likelihood function and estimation of the parameters on the basis of discrete data will be based on an approximation rather than an exact discretization. For example, the Euler method introduces discretization bias because it ignores the internal dynamics that can be excessively erratic. We view the approximation as a difference equation and note that the solution of the continuous time model does not satisfy this difference equation. The effectiveness of the approximation will depend on the rate at which the underlying process is sampled. We investigate how much it matters: can we get significantly different estimates of the same structural parameter when we use say hourly data as compared with using monthly data under given discretization? If yes, then that discretization when applied to a data set in hand, as is done in practice, cannot be said to give robust results. We compare numerically the application of methods by Yu and Phillips (2001), Shoji and Ozaki (1998) and Ait-Sahalia (2002) in the maximum likelihood estimation of the unrestricted interest rate model proposed by Chan et al. (1992). We find that reducing the sampling rate yield large biases in the estimation of the parameters. The AitSahalia method is shown to offer a good approximation and has the advantage of reducing some of the temporal aggregation bias.